SPRR Regressions
Steffen
12/5/2022
Steady State Databehandling
Silica Rejection
2D Multiple Linear Regression
We assume that pH is the major variable that controls silica rejection. A liniear regression between the two shows correlation to some degree.
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_SiO2 ~ pH, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.0437 -2.3231 0.2797 1.8346 4.7402
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -281.802 39.880 -7.066 5.88e-05 ***
## pH 32.493 4.155 7.820 2.65e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.25 on 9 degrees of freedom
## Multiple R-squared: 0.8717, Adjusted R-squared: 0.8574
## F-statistic: 61.15 on 1 and 9 DF, p-value: 2.654e-05## 3
## 0.4486406## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_SiO2 ~ Cl_SO4, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.0897 -5.9281 0.6586 6.9023 12.8352
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.856 15.261 1.694 0.124
## Cl_SO4 6.513 23.865 0.273 0.791
##
## Residual standard error: 9.037 on 9 degrees of freedom
## Multiple R-squared: 0.008207, Adjusted R-squared: -0.102
## F-statistic: 0.07447 on 1 and 9 DF, p-value: 0.7911## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_SiO2 ~ SiO2_mmol, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.168 -7.807 1.468 6.658 11.942
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 21.701 10.779 2.013 0.0749 .
## SiO2_mmol 5.092 6.447 0.790 0.4500
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.775 on 9 degrees of freedom
## Multiple R-squared: 0.06482, Adjusted R-squared: -0.03909
## F-statistic: 0.6238 on 1 and 9 DF, p-value: 0.45## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_SiO2 ~ pH + SiO2_mmol, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.3524 -1.3066 0.3034 0.6636 2.5001
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -296.952 22.680 -13.093 1.10e-06 ***
## pH 33.045 2.340 14.119 6.15e-07 ***
## SiO2_mmol 6.082 1.345 4.522 0.00195 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.828 on 8 degrees of freedom
## Multiple R-squared: 0.9639, Adjusted R-squared: 0.9549
## F-statistic: 106.9 on 2 and 8 DF, p-value: 1.695e-06
##
## Call:
## lm(formula = Rej_SiO2 ~ pH + Cl_anion, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9471 -2.3996 0.3955 1.7761 4.6450
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -282.799 43.397 -6.517 0.000185 ***
## pH 32.532 4.421 7.358 7.93e-05 ***
## Cl_anion 1.070 10.538 0.102 0.921625
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.445 on 8 degrees of freedom
## Multiple R-squared: 0.8719, Adjusted R-squared: 0.8398
## F-statistic: 27.22 on 2 and 8 DF, p-value: 0.0002696##
## Call:
## lm(formula = Rej_SiO2 ~ pH + Cl_SO4 + SiO2_mmol, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8346 -1.3148 -0.1641 0.8955 2.8465
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -298.998 22.996 -13.002 3.71e-06 ***
## pH 32.953 2.364 13.939 2.31e-06 ***
## Cl_SO4 4.512 4.882 0.924 0.38610
## SiO2_mmol 6.138 1.359 4.517 0.00274 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.845 on 7 degrees of freedom
## Multiple R-squared: 0.9678, Adjusted R-squared: 0.9541
## F-statistic: 70.23 on 3 and 7 DF, p-value: 1.371e-05Conclusion: Silica rejection can be predicted from pH and concentration of silica on these samples.
3D Representation
Silica as function of pH and silica concentration
Silica as function of pH and Cl/SO4 ratio
Silica as function of Cl/SO4 ratio and silica concentration
Cl rejection
2D Multiple Linear Regression
We assume that pH does not play a significant role in Cl rejection, at least not to the same degree as for SiO2. Instead the chemical environment i.e. which other species are present and in what amounts are theorized to have a greater influence.
We check if [Cl],[anion],[Cl]/[anion] has a significant role Anion fraktion udregneds udfra Cl/[SO4]+[SiO2_ladet]+[Cl]
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_Cl ~ Cl_SO4, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.864 -2.926 1.168 3.395 7.379
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.613 9.657 -0.685 0.511
## Cl_SO4 25.183 15.103 1.667 0.130
##
## Residual standard error: 5.719 on 9 degrees of freedom
## Multiple R-squared: 0.236, Adjusted R-squared: 0.1511
## F-statistic: 2.78 on 1 and 9 DF, p-value: 0.1298## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_Cl ~ Cl_anion, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.1764 -3.1815 0.2664 3.4872 7.4276
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -7.375 10.348 -0.713 0.494
## Cl_anion 28.524 17.523 1.628 0.138
##
## Residual standard error: 5.751 on 9 degrees of freedom
## Multiple R-squared: 0.2274, Adjusted R-squared: 0.1416
## F-statistic: 2.65 on 1 and 9 DF, p-value: 0.138## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_Cl ~ SO4_mmol, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.944 -2.997 1.103 3.457 6.812
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.228 3.944 3.861 0.00384 **
## SO4_mmol -1.919 1.136 -1.689 0.12543
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.701 on 9 degrees of freedom
## Multiple R-squared: 0.2407, Adjusted R-squared: 0.1564
## F-statistic: 2.854 on 1 and 9 DF, p-value: 0.1254## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_Cl ~ pH, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.882 -3.099 -0.138 4.707 7.573
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -87.996 73.442 -1.198 0.261
## pH 10.134 7.652 1.324 0.218
##
## Residual standard error: 5.986 on 9 degrees of freedom
## Multiple R-squared: 0.1631, Adjusted R-squared: 0.07009
## F-statistic: 1.754 on 1 and 9 DF, p-value: 0.2181## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = Rej_Cl ~ SiO2_mmol, data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.6785 -4.9632 0.7619 3.2193 11.3016
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.933 7.312 2.589 0.0292 *
## SiO2_mmol -5.986 4.373 -1.369 0.2043
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.953 on 9 degrees of freedom
## Multiple R-squared: 0.1723, Adjusted R-squared: 0.08029
## F-statistic: 1.873 on 1 and 9 DF, p-value: 0.2043##
## Call:
## lm(formula = Rej_Cl ~ SO4_mmol + total_sio2_ladet + Cl_mmol,
## data = Ion_data_frame)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.036 -2.382 1.022 2.983 7.982
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -54.537 57.374 -0.951 0.3735
## SO4_mmol -2.680 1.330 -2.015 0.0837 .
## total_sio2_ladet -3.580 8.933 -0.401 0.7005
## Cl_mmol 15.422 12.920 1.194 0.2715
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.844 on 7 degrees of freedom
## Multiple R-squared: 0.3795, Adjusted R-squared: 0.1135
## F-statistic: 1.427 on 3 and 7 DF, p-value: 0.3135
##
## Call:
## lm(formula = Rej_Cl ~ Cl_mmol * SO4_mmol * total_sio2_ladet,
## data = Ion_data_frame_both)
##
## Residuals:
## 1 2 3 4 5 6 7 8
## 2.28114 1.59695 0.52215 -2.88686 -2.26825 -4.42612 0.01679 -2.30095
## 9 10 11
## 4.56063 3.05730 -0.15277
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1044.434 941.557 -1.109 0.348
## Cl_mmol 232.414 200.726 1.158 0.331
## SO4_mmol 174.180 280.479 0.621 0.579
## total_sio2_ladet 620.850 1059.632 0.586 0.599
## Cl_mmol:SO4_mmol -39.706 58.412 -0.680 0.545
## Cl_mmol:total_sio2_ladet -144.214 223.126 -0.646 0.564
## SO4_mmol:total_sio2_ladet -18.938 404.575 -0.047 0.966
## Cl_mmol:SO4_mmol:total_sio2_ladet 8.229 82.808 0.099 0.927
##
## Residual standard error: 5.052 on 3 degrees of freedom
## Multiple R-squared: 0.8013, Adjusted R-squared: 0.3376
## F-statistic: 1.728 on 7 and 3 DF, p-value: 0.3526
##
## Call:
## lm(formula = Rej_Cl ~ SO4_mmol + Cl_mmol + SiO2_mmol, data = Ion_data_frame_both)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1347 -2.5512 0.2596 1.8279 5.7835
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -61.7651 37.9237 -1.629 0.1474
## SO4_mmol -2.6618 0.9648 -2.759 0.0281 *
## Cl_mmol 19.2005 8.4282 2.278 0.0568 .
## SiO2_mmol -8.1470 3.4806 -2.341 0.0518 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.427 on 7 degrees of freedom
## Multiple R-squared: 0.6439, Adjusted R-squared: 0.4913
## F-statistic: 4.22 on 3 and 7 DF, p-value: 0.05328
Conclusion: [Cl] not useful to predict Rejection of Cl neither is [anion] or [SO4] [Cl]/[Anion] has a weak correlation to Rejection R2 0.53 (significant P>0.003)
[Cl/Anion]+pH R2 0.63 Cl/Anion P>0.00225 **,pH P>0.07329 . .
Removing the high SO4 point shows how wack the correlations otherwise are.
3D Representation
All Data Points
Cl rejection as function of Cl/SO4 ratio pH
## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'Cl rejection as function of Cl/SO4 ratio and SiO2
## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'Cl rejection as function of sio2 and pH
## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'Removing of two outliers
Cl rejection as function of Cl/SO4 ratio pH
## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'Cl rejection as function of Cl/SO4 ratio and SiO2
## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'Cl rejection as function of sio2 and pH
## Warning: 'surface' objects don't have these attributes: 'mode', 'marker'
## Valid attributes include:
## 'type', 'visible', 'legendgroup', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'uirevision', 'z', 'x', 'y', 'text', 'hovertext', 'hovertemplate', 'connectgaps', 'surfacecolor', 'cauto', 'cmin', 'cmax', 'cmid', 'colorscale', 'autocolorscale', 'reversescale', 'showscale', 'colorbar', 'coloraxis', 'contours', 'hidesurface', 'lightposition', 'lighting', 'opacity', 'opacityscale', '_deprecated', 'hoverinfo', 'showlegend', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'zsrc', 'xsrc', 'ysrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'surfacecolorsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'